## Which properties of a random sequence are dynamically sensitive?(English)Zbl 1021.60055

A lot of classical results in probability theory, for instance the strong law of large numbers, the law of iterated logarithm, and so on, concern almost-sure properties of sequences $$\{X_n\}$$ of i.i.d. random variables.
Consider a sequence of i.i.d. random variables $$\{X_n\}$$ with state space $$S$$ and common law $$\nu$$, where each variable is replaced independently by an independent variable with the same law according to a Poisson clock, i.e., for each $$n\in\mathbb{N},$$ let $$X(t):=\{X_n(t)\},$$ $$t\geq 0$$, be an independent process which at rate 1 replaces its current value by an independent sample from $$\nu.$$ The distribution of $$X(t)$$ is $$\mu=\nu^{\mathbb{N}}$$ for every $$t\geq 0,$$ thus any Borel set $$A\subset S^{\mathbb{N}}$$ such that $$\mu(A)=1$$ satisfies for all $$t\geq 0,$$ $$P\{X(t)\in A\}=1$$ and $$P\{X(t)\in A\text{ for Lebesgue-a.e. }t\}=1.$$ The almost-sure properties of $$\{X_n\}$$, i.e., the subsets $$A$$ of $$S^{\mathbb{N}}$$ with $$\mu(A)=1$$ are classified by the authors as (dynamically) stable or sensitive according to whether or not the property: for all $$t\geq 0,$$ $$P\{X(t)\in A\text{ for Lebesgue-a.e. }t\}=1$$ can be strengthened to $$P\{X(t)\in A\text{ for all }t\}=1$$.
The authors prove that the law of large numbers and the law of iterated logarithm are dynamically stable while run tests are dynamically sensitive. They obtain multi-fractal analysis of exceptional times for run lengths and for prediction.
For the simple random walk in $$\mathbb{Z}^d$$ the authors prove that the transience is dynamically stable for $$d\geq 5,$$ and dynamically sensitive for $$d=3,4.$$ Moreover, for $$d=3,4$$ the nonempty random set of exceptional times $$t$$ where the random walk is recurrent has Hausdorff dimension $$(4-d)/ 2$$ a.s.

### MSC:

 60J25 Continuous-time Markov processes on general state spaces 28A80 Fractals 60F15 Strong limit theorems 28A78 Hausdorff and packing measures
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### References:

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